LMFDB - Embedded newform 784.6.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,6,Mod(1,784)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("784.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$125.740914733$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2732674592.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 113x^{2} + 882 $ x^4 - 113*x^2 + 882
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 392)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.90430$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-28.9227 q^{3} -63.5336 q^{5} +593.520 q^{9} +O(q^{10})$	$q-28.9227 q^{3} -63.5336 q^{5} +593.520 q^{9} +592.520 q^{11} +433.359 q^{13} +1837.56 q^{15} +32.2049 q^{17} +2712.56 q^{19} +3342.60 q^{23} +911.520 q^{25} -10138.0 q^{27} +8378.33 q^{29} +3301.03 q^{31} -17137.3 q^{33} -812.959 q^{37} -12533.9 q^{39} -8718.28 q^{41} +9723.64 q^{43} -37708.5 q^{45} +20235.8 q^{47} -931.451 q^{51} -9249.61 q^{53} -37645.0 q^{55} -78454.5 q^{57} +4039.25 q^{59} +10891.2 q^{61} -27532.8 q^{65} +52765.6 q^{67} -96677.0 q^{69} -52278.1 q^{71} -50714.2 q^{73} -26363.6 q^{75} +79876.5 q^{79} +148992. q^{81} -46491.8 q^{83} -2046.09 q^{85} -242324. q^{87} +148975. q^{89} -95474.7 q^{93} -172339. q^{95} -29601.7 q^{97} +351673. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 836 q^{9}+O(q^{10}) $ 4 * q + 836 * q^9	$ 4 q + 836 q^{9} + 832 q^{11} + 2736 q^{15} + 5680 q^{23} + 2108 q^{25} + 8904 q^{29} - 6328 q^{37} - 20912 q^{39} + 28128 q^{43} + 30112 q^{51} + 9144 q^{53} - 164624 q^{57} - 83984 q^{65} + 52640 q^{67} + 21600 q^{71} + 282592 q^{79} + 326804 q^{81} - 226592 q^{85} - 255776 q^{93} - 290992 q^{95} + 765312 q^{99}+O(q^{100}) $ 4 * q + 836 * q^9 + 832 * q^11 + 2736 * q^15 + 5680 * q^23 + 2108 * q^25 + 8904 * q^29 - 6328 * q^37 - 20912 * q^39 + 28128 * q^43 + 30112 * q^51 + 9144 * q^53 - 164624 * q^57 - 83984 * q^65 + 52640 * q^67 + 21600 * q^71 + 282592 * q^79 + 326804 * q^81 - 226592 * q^85 - 255776 * q^93 - 290992 * q^95 + 765312 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−28.9227	−1.85539	−0.927695	−	0.373339i	$-0.878213\pi$
$3$	−28.9227	−1.85539	−0.927695	+	0.373339i	$0.878213\pi$
$4$	0	0
$5$	−63.5336	−1.13652	−0.568262	−	0.822848i	$-0.692384\pi$
$5$	−63.5336	−1.13652	−0.568262	+	0.822848i	$0.692384\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	593.520	2.44247
$10$	0	0
$11$	592.520	1.47646	0.738230	−	0.674549i	$-0.235662\pi$
$11$	592.520	1.47646	0.738230	+	0.674549i	$0.235662\pi$
$12$	0	0
$13$	433.359	0.711196	0.355598	−	0.934639i	$-0.384277\pi$
$13$	433.359	0.711196	0.355598	+	0.934639i	$0.384277\pi$
$14$	0	0
$15$	1837.56	2.10869
$16$	0	0
$17$	32.2049	0.0270271	0.0135135	−	0.999909i	$-0.495698\pi$
$17$	32.2049	0.0270271	0.0135135	+	0.999909i	$0.495698\pi$
$18$	0	0
$19$	2712.56	1.72383	0.861917	−	0.507049i	$-0.169264\pi$
$19$	2712.56	1.72383	0.861917	+	0.507049i	$0.169264\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3342.60	1.31754	0.658772	−	0.752343i	$-0.271076\pi$
$23$	3342.60	1.31754	0.658772	+	0.752343i	$0.271076\pi$
$24$	0	0
$25$	911.520	0.291687
$26$	0	0
$27$	−10138.0	−2.67635
$28$	0	0
$29$	8378.33	1.84996	0.924980	−	0.380016i	$-0.124081\pi$
$29$	8378.33	1.84996	0.924980	+	0.380016i	$0.124081\pi$
$30$	0	0
$31$	3301.03	0.616944	0.308472	−	0.951233i	$-0.400182\pi$
$31$	3301.03	0.616944	0.308472	+	0.951233i	$0.400182\pi$
$32$	0	0
$33$	−17137.3	−2.73941
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−812.959	−0.0976257	−0.0488129	−	0.998808i	$-0.515544\pi$
$37$	−812.959	−0.0976257	−0.0488129	+	0.998808i	$0.515544\pi$
$38$	0	0
$39$	−12533.9	−1.31955
$40$	0	0
$41$	−8718.28	−0.809974	−0.404987	−	0.914322i	$-0.632724\pi$
$41$	−8718.28	−0.809974	−0.404987	+	0.914322i	$0.632724\pi$
$42$	0	0
$43$	9723.64	0.801970	0.400985	−	0.916085i	$-0.368668\pi$
$43$	9723.64	0.801970	0.400985	+	0.916085i	$0.368668\pi$
$44$	0	0
$45$	−37708.5	−2.77593
$46$	0	0
$47$	20235.8	1.33621	0.668106	−	0.744066i	$-0.267105\pi$
$47$	20235.8	1.33621	0.668106	+	0.744066i	$0.267105\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−931.451	−0.0501458
$52$	0	0
$53$	−9249.61	−0.452308	−0.226154	−	0.974092i	$-0.572615\pi$
$53$	−9249.61	−0.452308	−0.226154	+	0.974092i	$0.572615\pi$
$54$	0	0
$55$	−37645.0	−1.67803
$56$	0	0
$57$	−78454.5	−3.19838
$58$	0	0
$59$	4039.25	0.151067	0.0755337	−	0.997143i	$-0.475934\pi$
$59$	4039.25	0.151067	0.0755337	+	0.997143i	$0.475934\pi$
$60$	0	0
$61$	10891.2	0.374758	0.187379	−	0.982288i	$-0.440001\pi$
$61$	10891.2	0.374758	0.187379	+	0.982288i	$0.440001\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−27532.8	−0.808291
$66$	0	0
$67$	52765.6	1.43603	0.718016	−	0.696027i	$-0.245051\pi$
$67$	52765.6	1.43603	0.718016	+	0.696027i	$0.245051\pi$
$68$	0	0
$69$	−96677.0	−2.44456
$70$	0	0
$71$	−52278.1	−1.23076	−0.615380	−	0.788230i	$-0.710998\pi$
$71$	−52278.1	−1.23076	−0.615380	+	0.788230i	$0.710998\pi$
$72$	0	0
$73$	−50714.2	−1.11384	−0.556919	−	0.830567i	$-0.688017\pi$
$73$	−50714.2	−1.11384	−0.556919	+	0.830567i	$0.688017\pi$
$74$	0	0
$75$	−26363.6	−0.541192
$76$	0	0
$77$	0	0
$78$	0	0
$79$	79876.5	1.43996	0.719981	−	0.693993i	$-0.244150\pi$
$79$	79876.5	1.43996	0.719981	+	0.693993i	$0.244150\pi$
$80$	0	0
$81$	148992.	2.52319
$82$	0	0
$83$	−46491.8	−0.740767	−0.370383	−	0.928879i	$-0.620774\pi$
$83$	−46491.8	−0.740767	−0.370383	+	0.928879i	$0.620774\pi$
$84$	0	0
$85$	−2046.09	−0.0307169
$86$	0	0
$87$	−242324.	−3.43240
$88$	0	0
$89$	148975.	1.99360	0.996800	−	0.0799329i	$-0.0254706\pi$
$89$	148975.	1.99360	0.996800	+	0.0799329i	$0.0254706\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−95474.7	−1.14467
$94$	0	0
$95$	−172339.	−1.95918
$96$	0	0
$97$	−29601.7	−0.319439	−0.159719	−	0.987162i	$-0.551059\pi$
$97$	−29601.7	−0.319439	−0.159719	+	0.987162i	$0.551059\pi$
$98$	0	0
$99$	351673.	3.60621
$100$	0	0
$101$	106123.	1.03515	0.517577	−	0.855636i	$-0.326834\pi$
$101$	106123.	1.03515	0.517577	+	0.855636i	$0.326834\pi$
$102$	0	0
$103$	26722.6	0.248190	0.124095	−	0.992270i	$-0.460397\pi$
$103$	26722.6	0.248190	0.124095	+	0.992270i	$0.460397\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	108549.	0.916573	0.458287	−	0.888804i	$-0.348463\pi$
$107$	108549.	0.916573	0.458287	+	0.888804i	$0.348463\pi$
$108$	0	0
$109$	−76781.6	−0.619000	−0.309500	−	0.950899i	$-0.600162\pi$
$109$	−76781.6	−0.619000	−0.309500	+	0.950899i	$0.600162\pi$
$110$	0	0
$111$	23512.9	0.181134
$112$	0	0
$113$	−89821.9	−0.661738	−0.330869	−	0.943677i	$-0.607342\pi$
$113$	−89821.9	−0.661738	−0.330869	+	0.943677i	$0.607342\pi$
$114$	0	0
$115$	−212368.	−1.49742
$116$	0	0
$117$	257207.	1.73708
$118$	0	0
$119$	0	0
$120$	0	0
$121$	190030.	1.17993
$122$	0	0
$123$	252156.	1.50282
$124$	0	0
$125$	140630.	0.805015
$126$	0	0
$127$	48119.5	0.264735	0.132368	−	0.991201i	$-0.457742\pi$
$127$	48119.5	0.264735	0.132368	+	0.991201i	$0.457742\pi$
$128$	0	0
$129$	−281234.	−1.48797
$130$	0	0
$131$	−6888.93	−0.0350730	−0.0175365	−	0.999846i	$-0.505582\pi$
$131$	−6888.93	−0.0350730	−0.0175365	+	0.999846i	$0.505582\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	644103.	3.04173
$136$	0	0
$137$	103794.	0.472467	0.236233	−	0.971696i	$-0.424087\pi$
$137$	103794.	0.472467	0.236233	+	0.971696i	$0.424087\pi$
$138$	0	0
$139$	325743.	1.43001	0.715003	−	0.699121i	$-0.246425\pi$
$139$	325743.	1.43001	0.715003	+	0.699121i	$0.246425\pi$
$140$	0	0
$141$	−585273.	−2.47920
$142$	0	0
$143$	256774.	1.05005
$144$	0	0
$145$	−532305.	−2.10252
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−109769.	−0.405055	−0.202528	−	0.979277i	$-0.564916\pi$
$149$	−109769.	−0.405055	−0.202528	+	0.979277i	$0.564916\pi$
$150$	0	0
$151$	−312238.	−1.11441	−0.557204	−	0.830376i	$-0.688126\pi$
$151$	−312238.	−1.11441	−0.557204	+	0.830376i	$0.688126\pi$
$152$	0	0
$153$	19114.2	0.0660129
$154$	0	0
$155$	−209727.	−0.701171
$156$	0	0
$157$	120613.	0.390520	0.195260	−	0.980751i	$-0.437445\pi$
$157$	120613.	0.390520	0.195260	+	0.980751i	$0.437445\pi$
$158$	0	0
$159$	267523.	0.839207
$160$	0	0
$161$	0	0
$162$	0	0
$163$	176764.	0.521104	0.260552	−	0.965460i	$-0.416095\pi$
$163$	176764.	0.521104	0.260552	+	0.965460i	$0.416095\pi$
$164$	0	0
$165$	1.08879e6	3.11340
$166$	0	0
$167$	−487840.	−1.35359	−0.676794	−	0.736172i	$-0.736631\pi$
$167$	−487840.	−1.35359	−0.676794	+	0.736172i	$0.736631\pi$
$168$	0	0
$169$	−183493.	−0.494201
$170$	0	0
$171$	1.60996e6	4.21042
$172$	0	0
$173$	−77727.6	−0.197451	−0.0987257	−	0.995115i	$-0.531477\pi$
$173$	−77727.6	−0.197451	−0.0987257	+	0.995115i	$0.531477\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−116826.	−0.280289
$178$	0	0
$179$	307721.	0.717834	0.358917	−	0.933369i	$-0.383146\pi$
$179$	307721.	0.717834	0.358917	+	0.933369i	$0.383146\pi$
$180$	0	0
$181$	210791.	0.478251	0.239126	−	0.970989i	$-0.423139\pi$
$181$	210791.	0.478251	0.239126	+	0.970989i	$0.423139\pi$
$182$	0	0
$183$	−315002.	−0.695322
$184$	0	0
$185$	51650.2	0.110954
$186$	0	0
$187$	19082.0	0.0399044
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−593394.	−1.17695	−0.588477	−	0.808514i	$-0.700272\pi$
$191$	−593394.	−1.17695	−0.588477	+	0.808514i	$0.700272\pi$
$192$	0	0
$193$	−369858.	−0.714729	−0.357365	−	0.933965i	$-0.616325\pi$
$193$	−369858.	−0.714729	−0.357365	+	0.933965i	$0.616325\pi$
$194$	0	0
$195$	796323.	1.49969
$196$	0	0
$197$	−458329.	−0.841419	−0.420709	−	0.907196i	$-0.638219\pi$
$197$	−458329.	−0.841419	−0.420709	+	0.907196i	$0.638219\pi$
$198$	0	0
$199$	−444199.	−0.795142	−0.397571	−	0.917571i	$-0.630147\pi$
$199$	−444199.	−0.795142	−0.397571	+	0.917571i	$0.630147\pi$
$200$	0	0
$201$	−1.52612e6	−2.66440
$202$	0	0
$203$	0	0
$204$	0	0
$205$	553904.	0.920555
$206$	0	0
$207$	1.98390e6	3.21806
$208$	0	0
$209$	1.60725e6	2.54517
$210$	0	0
$211$	327421.	0.506291	0.253146	−	0.967428i	$-0.418535\pi$
$211$	327421.	0.506291	0.253146	+	0.967428i	$0.418535\pi$
$212$	0	0
$213$	1.51202e6	2.28354
$214$	0	0
$215$	−617778.	−0.911458
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.46679e6	2.06660
$220$	0	0
$221$	13956.3	0.0192215
$222$	0	0
$223$	−449149.	−0.604823	−0.302411	−	0.953177i	$-0.597792\pi$
$223$	−449149.	−0.604823	−0.302411	+	0.953177i	$0.597792\pi$
$224$	0	0
$225$	541006.	0.712436
$226$	0	0
$227$	−36225.2	−0.0466602	−0.0233301	−	0.999728i	$-0.507427\pi$
$227$	−36225.2	−0.0466602	−0.0233301	+	0.999728i	$0.507427\pi$
$228$	0	0
$229$	807858.	1.01800	0.508998	−	0.860768i	$-0.330016\pi$
$229$	807858.	1.01800	0.508998	+	0.860768i	$0.330016\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.21525e6	1.46648	0.733242	−	0.679968i	$-0.238006\pi$
$233$	1.21525e6	1.46648	0.733242	+	0.679968i	$0.238006\pi$
$234$	0	0
$235$	−1.28565e6	−1.51864
$236$	0	0
$237$	−2.31024e6	−2.67169
$238$	0	0
$239$	203963.	0.230970	0.115485	−	0.993309i	$-0.463158\pi$
$239$	203963.	0.230970	0.115485	+	0.993309i	$0.463158\pi$
$240$	0	0
$241$	650170.	0.721082	0.360541	−	0.932743i	$-0.382592\pi$
$241$	650170.	0.721082	0.360541	+	0.932743i	$0.382592\pi$
$242$	0	0
$243$	−1.84572e6	−2.00516
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.17551e6	1.22598
$248$	0	0
$249$	1.34467e6	1.37441
$250$	0	0
$251$	69444.4	0.0695750	0.0347875	−	0.999395i	$-0.488925\pi$
$251$	69444.4	0.0695750	0.0347875	+	0.999395i	$0.488925\pi$
$252$	0	0
$253$	1.98056e6	1.94530
$254$	0	0
$255$	59178.4	0.0569919
$256$	0	0
$257$	332991.	0.314485	0.157242	−	0.987560i	$-0.449740\pi$
$257$	332991.	0.314485	0.157242	+	0.987560i	$0.449740\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	4.97271e6	4.51847
$262$	0	0
$263$	896118.	0.798870	0.399435	−	0.916762i	$-0.369206\pi$
$263$	896118.	0.798870	0.399435	+	0.916762i	$0.369206\pi$
$264$	0	0
$265$	587661.	0.514058
$266$	0	0
$267$	−4.30875e6	−3.69891
$268$	0	0
$269$	1.70580e6	1.43730	0.718652	−	0.695370i	$-0.244760\pi$
$269$	1.70580e6	1.43730	0.718652	+	0.695370i	$0.244760\pi$
$270$	0	0
$271$	69736.1	0.0576812	0.0288406	−	0.999584i	$-0.490818\pi$
$271$	69736.1	0.0576812	0.0288406	+	0.999584i	$0.490818\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	540095.	0.430663
$276$	0	0
$277$	−559541.	−0.438160	−0.219080	−	0.975707i	$-0.570305\pi$
$277$	−559541.	−0.438160	−0.219080	+	0.975707i	$0.570305\pi$
$278$	0	0
$279$	1.95923e6	1.50687
$280$	0	0
$281$	−837805.	−0.632961	−0.316481	−	0.948599i	$-0.602501\pi$
$281$	−837805.	−0.632961	−0.316481	+	0.948599i	$0.602501\pi$
$282$	0	0
$283$	−1.83538e6	−1.36226	−0.681131	−	0.732162i	$-0.738512\pi$
$283$	−1.83538e6	−1.36226	−0.681131	+	0.732162i	$0.738512\pi$
$284$	0	0
$285$	4.98450e6	3.63504
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.41882e6	−0.999270
$290$	0	0
$291$	856161.	0.592684
$292$	0	0
$293$	245011.	0.166731	0.0833655	−	0.996519i	$-0.473433\pi$
$293$	245011.	0.166731	0.0833655	+	0.996519i	$0.473433\pi$
$294$	0	0
$295$	−256628.	−0.171692
$296$	0	0
$297$	−6.00696e6	−3.95152
$298$	0	0
$299$	1.44855e6	0.937032
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−3.06935e6	−1.92062
$304$	0	0
$305$	−691957.	−0.425922
$306$	0	0
$307$	396561.	0.240140	0.120070	−	0.992765i	$-0.461688\pi$
$307$	396561.	0.240140	0.120070	+	0.992765i	$0.461688\pi$
$308$	0	0
$309$	−772888.	−0.460490
$310$	0	0
$311$	−2.14081e6	−1.25509	−0.627547	−	0.778579i	$-0.715941\pi$
$311$	−2.14081e6	−1.25509	−0.627547	+	0.778579i	$0.715941\pi$
$312$	0	0
$313$	−3.24441e6	−1.87187	−0.935934	−	0.352175i	$-0.885442\pi$
$313$	−3.24441e6	−1.87187	−0.935934	+	0.352175i	$0.885442\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.63757e6	−1.47420	−0.737100	−	0.675784i	$-0.763805\pi$
$317$	−2.63757e6	−1.47420	−0.737100	+	0.675784i	$0.763805\pi$
$318$	0	0
$319$	4.96433e6	2.73139
$320$	0	0
$321$	−3.13953e6	−1.70060
$322$	0	0
$323$	87357.7	0.0465902
$324$	0	0
$325$	395015.	0.207446
$326$	0	0
$327$	2.22073e6	1.14849
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.83433e6	−0.920251	−0.460126	−	0.887854i	$-0.652196\pi$
$331$	−1.83433e6	−0.920251	−0.460126	+	0.887854i	$0.652196\pi$
$332$	0	0
$333$	−482508.	−0.238448
$334$	0	0
$335$	−3.35239e6	−1.63208
$336$	0	0
$337$	433695.	0.208022	0.104011	−	0.994576i	$-0.466832\pi$
$337$	433695.	0.208022	0.104011	+	0.994576i	$0.466832\pi$
$338$	0	0
$339$	2.59789e6	1.22778
$340$	0	0
$341$	1.95593e6	0.910893
$342$	0	0
$343$	0	0
$344$	0	0
$345$	6.14224e6	2.77830
$346$	0	0
$347$	703899.	0.313825	0.156912	−	0.987613i	$-0.449846\pi$
$347$	703899.	0.313825	0.156912	+	0.987613i	$0.449846\pi$
$348$	0	0
$349$	2.26554e6	0.995653	0.497826	−	0.867277i	$-0.334132\pi$
$349$	2.26554e6	0.995653	0.497826	+	0.867277i	$0.334132\pi$
$350$	0	0
$351$	−4.39338e6	−1.90341
$352$	0	0
$353$	−3.84527e6	−1.64244	−0.821220	−	0.570611i	$-0.806706\pi$
$353$	−3.84527e6	−1.64244	−0.821220	+	0.570611i	$0.806706\pi$
$354$	0	0
$355$	3.32142e6	1.39879
$356$	0	0
$357$	0	0
$358$	0	0
$359$	263792.	0.108025	0.0540126	−	0.998540i	$-0.482799\pi$
$359$	263792.	0.108025	0.0540126	+	0.998540i	$0.482799\pi$
$360$	0	0
$361$	4.88189e6	1.97160
$362$	0	0
$363$	−5.49616e6	−2.18924
$364$	0	0
$365$	3.22205e6	1.26590
$366$	0	0
$367$	−3.09852e6	−1.20085	−0.600426	−	0.799680i	$-0.705002\pi$
$367$	−3.09852e6	−1.20085	−0.600426	+	0.799680i	$0.705002\pi$
$368$	0	0
$369$	−5.17448e6	−1.97834
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−460228.	−0.171278	−0.0856388	−	0.996326i	$-0.527293\pi$
$373$	−460228.	−0.171278	−0.0856388	+	0.996326i	$0.527293\pi$
$374$	0	0
$375$	−4.06740e6	−1.49362
$376$	0	0
$377$	3.63082e6	1.31568
$378$	0	0
$379$	951319.	0.340195	0.170098	−	0.985427i	$-0.445592\pi$
$379$	951319.	0.340195	0.170098	+	0.985427i	$0.445592\pi$
$380$	0	0
$381$	−1.39174e6	−0.491187
$382$	0	0
$383$	3.35711e6	1.16942	0.584708	−	0.811244i	$-0.301209\pi$
$383$	3.35711e6	1.16942	0.584708	+	0.811244i	$0.301209\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.77118e6	1.95879
$388$	0	0
$389$	3.31496e6	1.11072	0.555359	−	0.831611i	$-0.312581\pi$
$389$	3.31496e6	1.11072	0.555359	+	0.831611i	$0.312581\pi$
$390$	0	0
$391$	107648.	0.0356094
$392$	0	0
$393$	199246.	0.0650741
$394$	0	0
$395$	−5.07484e6	−1.63655
$396$	0	0
$397$	−2.00849e6	−0.639576	−0.319788	−	0.947489i	$-0.603612\pi$
$397$	−2.00849e6	−0.639576	−0.319788	+	0.947489i	$0.603612\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.76149e6	0.857595	0.428797	−	0.903401i	$-0.358937\pi$
$401$	2.76149e6	0.857595	0.428797	+	0.903401i	$0.358937\pi$
$402$	0	0
$403$	1.43053e6	0.438768
$404$	0	0
$405$	−9.46601e6	−2.86767
$406$	0	0
$407$	−481695.	−0.144140
$408$	0	0
$409$	−2.93620e6	−0.867916	−0.433958	−	0.900933i	$-0.642883\pi$
$409$	−2.93620e6	−0.867916	−0.433958	+	0.900933i	$0.642883\pi$
$410$	0	0
$411$	−3.00200e6	−0.876610
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2.95380e6	0.841899
$416$	0	0
$417$	−9.42135e6	−2.65322
$418$	0	0
$419$	−355495.	−0.0989233	−0.0494617	−	0.998776i	$-0.515751\pi$
$419$	−355495.	−0.0989233	−0.0494617	+	0.998776i	$0.515751\pi$
$420$	0	0
$421$	−4.22340e6	−1.16133	−0.580666	−	0.814142i	$-0.697208\pi$
$421$	−4.22340e6	−1.16133	−0.580666	+	0.814142i	$0.697208\pi$
$422$	0	0
$423$	1.20104e7	3.26366
$424$	0	0
$425$	29355.4	0.00788344
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−7.42659e6	−1.94826
$430$	0	0
$431$	−5.75307e6	−1.49179	−0.745893	−	0.666066i	$-0.767977\pi$
$431$	−5.75307e6	−1.49179	−0.745893	+	0.666066i	$0.767977\pi$
$432$	0	0
$433$	6.17411e6	1.58254	0.791269	−	0.611468i	$-0.209421\pi$
$433$	6.17411e6	1.58254	0.791269	+	0.611468i	$0.209421\pi$
$434$	0	0
$435$	1.53957e7	3.90100
$436$	0	0
$437$	9.06701e6	2.27123
$438$	0	0
$439$	−6.07395e6	−1.50422	−0.752108	−	0.659040i	$-0.770963\pi$
$439$	−6.07395e6	−1.50422	−0.752108	+	0.659040i	$0.770963\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.98673e6	−0.965177	−0.482589	−	0.875847i	$-0.660303\pi$
$443$	−3.98673e6	−0.965177	−0.482589	+	0.875847i	$0.660303\pi$
$444$	0	0
$445$	−9.46492e6	−2.26577
$446$	0	0
$447$	3.17481e6	0.751535
$448$	0	0
$449$	3.65019e6	0.854475	0.427238	−	0.904139i	$-0.359487\pi$
$449$	3.65019e6	0.854475	0.427238	+	0.904139i	$0.359487\pi$
$450$	0	0
$451$	−5.16576e6	−1.19589
$452$	0	0
$453$	9.03077e6	2.06766
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−654800.	−0.146662	−0.0733311	−	0.997308i	$-0.523363\pi$
$457$	−654800.	−0.146662	−0.0733311	+	0.997308i	$0.523363\pi$
$458$	0	0
$459$	−326493.	−0.0723338
$460$	0	0
$461$	−5.72912e6	−1.25555	−0.627777	−	0.778393i	$-0.716035\pi$
$461$	−5.72912e6	−1.25555	−0.627777	+	0.778393i	$0.716035\pi$
$462$	0	0
$463$	−6.03283e6	−1.30788	−0.653941	−	0.756545i	$-0.726886\pi$
$463$	−6.03283e6	−1.30788	−0.653941	+	0.756545i	$0.726886\pi$
$464$	0	0
$465$	6.06585e6	1.30095
$466$	0	0
$467$	3.23056e6	0.685465	0.342732	−	0.939433i	$-0.388648\pi$
$467$	3.23056e6	0.685465	0.342732	+	0.939433i	$0.388648\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3.48844e6	−0.724567
$472$	0	0
$473$	5.76146e6	1.18408
$474$	0	0
$475$	2.47255e6	0.502819
$476$	0	0
$477$	−5.48984e6	−1.10475
$478$	0	0
$479$	4.22571e6	0.841513	0.420756	−	0.907174i	$-0.361765\pi$
$479$	4.22571e6	0.841513	0.420756	+	0.907174i	$0.361765\pi$
$480$	0	0
$481$	−352303.	−0.0694310
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.88070e6	0.363050
$486$	0	0
$487$	−3.71427e6	−0.709662	−0.354831	−	0.934930i	$-0.615462\pi$
$487$	−3.71427e6	−0.709662	−0.354831	+	0.934930i	$0.615462\pi$
$488$	0	0
$489$	−5.11248e6	−0.966851
$490$	0	0
$491$	6.98790e6	1.30811	0.654053	−	0.756449i	$-0.273067\pi$
$491$	6.98790e6	1.30811	0.654053	+	0.756449i	$0.273067\pi$
$492$	0	0
$493$	269823.	0.0499990
$494$	0	0
$495$	−2.23431e7	−4.09854
$496$	0	0
$497$	0	0
$498$	0	0
$499$	366427.	0.0658773	0.0329387	−	0.999457i	$-0.489513\pi$
$499$	366427.	0.0658773	0.0329387	+	0.999457i	$0.489513\pi$
$500$	0	0
$501$	1.41096e7	2.51143
$502$	0	0
$503$	9.94067e6	1.75185	0.875923	−	0.482452i	$-0.160254\pi$
$503$	9.94067e6	1.75185	0.875923	+	0.482452i	$0.160254\pi$
$504$	0	0
$505$	−6.74237e6	−1.17648
$506$	0	0
$507$	5.30711e6	0.916935
$508$	0	0
$509$	−2.23471e6	−0.382320	−0.191160	−	0.981559i	$-0.561225\pi$
$509$	−2.23471e6	−0.382320	−0.191160	+	0.981559i	$0.561225\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.74999e7	−4.61358
$514$	0	0
$515$	−1.69778e6	−0.282074
$516$	0	0
$517$	1.19901e7	1.97286
$518$	0	0
$519$	2.24809e6	0.366349
$520$	0	0
$521$	9.40325e6	1.51769	0.758846	−	0.651271i	$-0.225764\pi$
$521$	9.40325e6	1.51769	0.758846	+	0.651271i	$0.225764\pi$
$522$	0	0
$523$	−7.59427e6	−1.21404	−0.607018	−	0.794688i	$-0.707634\pi$
$523$	−7.59427e6	−1.21404	−0.607018	+	0.794688i	$0.707634\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	106309.	0.0166742
$528$	0	0
$529$	4.73665e6	0.735922
$530$	0	0
$531$	2.39738e6	0.368978
$532$	0	0
$533$	−3.77814e6	−0.576050
$534$	0	0
$535$	−6.89652e6	−1.04171
$536$	0	0
$537$	−8.90010e6	−1.33186
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8.38619e6	−1.23189	−0.615945	−	0.787789i	$-0.711226\pi$
$541$	−8.38619e6	−1.23189	−0.615945	+	0.787789i	$0.711226\pi$
$542$	0	0
$543$	−6.09664e6	−0.887342
$544$	0	0
$545$	4.87821e6	0.703509
$546$	0	0
$547$	−5.66890e6	−0.810084	−0.405042	−	0.914298i	$-0.632743\pi$
$547$	−5.66890e6	−0.810084	−0.405042	+	0.914298i	$0.632743\pi$
$548$	0	0
$549$	6.46415e6	0.915336
$550$	0	0
$551$	2.27267e7	3.18902
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.49386e6	−0.205863
$556$	0	0
$557$	−7.27409e6	−0.993438	−0.496719	−	0.867911i	$-0.665462\pi$
$557$	−7.27409e6	−0.993438	−0.496719	+	0.867911i	$0.665462\pi$
$558$	0	0
$559$	4.21383e6	0.570357
$560$	0	0
$561$	−551904.	−0.0740382
$562$	0	0
$563$	−1.69351e6	−0.225173	−0.112586	−	0.993642i	$-0.535914\pi$
$563$	−1.69351e6	−0.225173	−0.112586	+	0.993642i	$0.535914\pi$
$564$	0	0
$565$	5.70671e6	0.752081
$566$	0	0
$567$	0	0
$568$	0	0
$569$	868712.	0.112485	0.0562426	−	0.998417i	$-0.482088\pi$
$569$	868712.	0.112485	0.0562426	+	0.998417i	$0.482088\pi$
$570$	0	0
$571$	2.85740e6	0.366759	0.183380	−	0.983042i	$-0.441296\pi$
$571$	2.85740e6	0.366759	0.183380	+	0.983042i	$0.441296\pi$
$572$	0	0
$573$	1.71625e7	2.18371
$574$	0	0
$575$	3.04685e6	0.384310
$576$	0	0
$577$	−764271.	−0.0955670	−0.0477835	−	0.998858i	$-0.515216\pi$
$577$	−764271.	−0.0955670	−0.0477835	+	0.998858i	$0.515216\pi$
$578$	0	0
$579$	1.06973e7	1.32610
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−5.48059e6	−0.667814
$584$	0	0
$585$	−1.63413e7	−1.97423
$586$	0	0
$587$	3.53684e6	0.423663	0.211832	−	0.977306i	$-0.432057\pi$
$587$	3.53684e6	0.423663	0.211832	+	0.977306i	$0.432057\pi$
$588$	0	0
$589$	8.95425e6	1.06351
$590$	0	0
$591$	1.32561e7	1.56116
$592$	0	0
$593$	1.39698e7	1.63137	0.815686	−	0.578495i	$-0.196360\pi$
$593$	1.39698e7	1.63137	0.815686	+	0.578495i	$0.196360\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.28474e7	1.47530
$598$	0	0
$599$	7.65719e6	0.871972	0.435986	−	0.899954i	$-0.356400\pi$
$599$	7.65719e6	0.871972	0.435986	+	0.899954i	$0.356400\pi$
$600$	0	0
$601$	5.93011e6	0.669694	0.334847	−	0.942272i	$-0.391315\pi$
$601$	5.93011e6	0.669694	0.334847	+	0.942272i	$0.391315\pi$
$602$	0	0
$603$	3.13175e7	3.50747
$604$	0	0
$605$	−1.20733e7	−1.34102
$606$	0	0
$607$	7.71203e6	0.849565	0.424783	−	0.905295i	$-0.360350\pi$
$607$	7.71203e6	0.849565	0.424783	+	0.905295i	$0.360350\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.76936e6	0.950309
$612$	0	0
$613$	−1.64407e7	−1.76713	−0.883564	−	0.468311i	$-0.844863\pi$
$613$	−1.64407e7	−1.76713	−0.883564	+	0.468311i	$0.844863\pi$
$614$	0	0
$615$	−1.60204e7	−1.70799
$616$	0	0
$617$	−1.11040e7	−1.17427	−0.587133	−	0.809491i	$-0.699743\pi$
$617$	−1.11040e7	−1.17427	−0.587133	+	0.809491i	$0.699743\pi$
$618$	0	0
$619$	8.44209e6	0.885571	0.442785	−	0.896628i	$-0.353990\pi$
$619$	8.44209e6	0.885571	0.442785	+	0.896628i	$0.353990\pi$
$620$	0	0
$621$	−3.38873e7	−3.52620
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.17833e7	−1.20661
$626$	0	0
$627$	−4.64859e7	−4.72229
$628$	0	0
$629$	−26181.2	−0.00263854
$630$	0	0
$631$	3.47137e6	0.347078	0.173539	−	0.984827i	$-0.444480\pi$
$631$	3.47137e6	0.347078	0.173539	+	0.984827i	$0.444480\pi$
$632$	0	0
$633$	−9.46988e6	−0.939367
$634$	0	0
$635$	−3.05721e6	−0.300878
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3.10281e7	−3.00610
$640$	0	0
$641$	−7.34206e6	−0.705786	−0.352893	−	0.935664i	$-0.614802\pi$
$641$	−7.34206e6	−0.705786	−0.352893	+	0.935664i	$0.614802\pi$
$642$	0	0
$643$	−1.28361e6	−0.122435	−0.0612177	−	0.998124i	$-0.519498\pi$
$643$	−1.28361e6	−0.122435	−0.0612177	+	0.998124i	$0.519498\pi$
$644$	0	0
$645$	1.78678e7	1.69111
$646$	0	0
$647$	−3.09922e6	−0.291066	−0.145533	−	0.989353i	$-0.546490\pi$
$647$	−3.09922e6	−0.291066	−0.145533	+	0.989353i	$0.546490\pi$
$648$	0	0
$649$	2.39334e6	0.223045
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.46219e6	−0.225964	−0.112982	−	0.993597i	$-0.536040\pi$
$653$	−2.46219e6	−0.225964	−0.112982	+	0.993597i	$0.536040\pi$
$654$	0	0
$655$	437678.	0.0398613
$656$	0	0
$657$	−3.00999e7	−2.72052
$658$	0	0
$659$	−6.60217e6	−0.592206	−0.296103	−	0.955156i	$-0.595687\pi$
$659$	−6.60217e6	−0.592206	−0.296103	+	0.955156i	$0.595687\pi$
$660$	0	0
$661$	−1.79222e7	−1.59547	−0.797733	−	0.603011i	$-0.793968\pi$
$661$	−1.79222e7	−1.59547	−0.797733	+	0.603011i	$0.793968\pi$
$662$	0	0
$663$	−403652.	−0.0356635
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.80054e7	2.43740
$668$	0	0
$669$	1.29906e7	1.12218
$670$	0	0
$671$	6.45326e6	0.553315
$672$	0	0
$673$	1.90502e7	1.62130	0.810648	−	0.585534i	$-0.199115\pi$
$673$	1.90502e7	1.62130	0.810648	+	0.585534i	$0.199115\pi$
$674$	0	0
$675$	−9.24098e6	−0.780654
$676$	0	0
$677$	−8.09514e6	−0.678817	−0.339408	−	0.940639i	$-0.610227\pi$
$677$	−8.09514e6	−0.678817	−0.339408	+	0.940639i	$0.610227\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.04773e6	0.0865728
$682$	0	0
$683$	−7.69330e6	−0.631046	−0.315523	−	0.948918i	$-0.602180\pi$
$683$	−7.69330e6	−0.631046	−0.315523	+	0.948918i	$0.602180\pi$
$684$	0	0
$685$	−6.59441e6	−0.536970
$686$	0	0
$687$	−2.33654e7	−1.88878
$688$	0	0
$689$	−4.00840e6	−0.321679
$690$	0	0
$691$	1.64579e7	1.31123	0.655616	−	0.755094i	$-0.272409\pi$
$691$	1.64579e7	1.31123	0.655616	+	0.755094i	$0.272409\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.06956e7	−1.62524
$696$	0	0
$697$	−280771.	−0.0218912
$698$	0	0
$699$	−3.51484e7	−2.72090
$700$	0	0
$701$	3.18690e6	0.244948	0.122474	−	0.992472i	$-0.460917\pi$
$701$	3.18690e6	0.244948	0.122474	+	0.992472i	$0.460917\pi$
$702$	0	0
$703$	−2.20520e6	−0.168291
$704$	0	0
$705$	3.71845e7	2.81767
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.60785e7	−1.20124	−0.600621	−	0.799534i	$-0.705080\pi$
$709$	−1.60785e7	−1.20124	−0.600621	+	0.799534i	$0.705080\pi$
$710$	0	0
$711$	4.74083e7	3.51707
$712$	0	0
$713$	1.10340e7	0.812851
$714$	0	0
$715$	−1.63138e7	−1.19341
$716$	0	0
$717$	−5.89915e6	−0.428540
$718$	0	0
$719$	−1.15621e7	−0.834096	−0.417048	−	0.908885i	$-0.636935\pi$
$719$	−1.15621e7	−0.834096	−0.417048	+	0.908885i	$0.636935\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.88047e7	−1.33789
$724$	0	0
$725$	7.63702e6	0.539608
$726$	0	0
$727$	2.06100e7	1.44624	0.723122	−	0.690720i	$-0.242706\pi$
$727$	2.06100e7	1.44624	0.723122	+	0.690720i	$0.242706\pi$
$728$	0	0
$729$	1.71780e7	1.19716
$730$	0	0
$731$	313149.	0.0216749
$732$	0	0
$733$	7.55344e6	0.519260	0.259630	−	0.965708i	$-0.416399\pi$
$733$	7.55344e6	0.519260	0.259630	+	0.965708i	$0.416399\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.12647e7	2.12024
$738$	0	0
$739$	1.50829e7	1.01595	0.507975	−	0.861372i	$-0.330394\pi$
$739$	1.50829e7	1.01595	0.507975	+	0.861372i	$0.330394\pi$
$740$	0	0
$741$	−3.39989e7	−2.27468
$742$	0	0
$743$	971810.	0.0645817	0.0322908	−	0.999479i	$-0.489720\pi$
$743$	971810.	0.0645817	0.0322908	+	0.999479i	$0.489720\pi$
$744$	0	0
$745$	6.97402e6	0.460355
$746$	0	0
$747$	−2.75939e7	−1.80930
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.59178e6	0.102987	0.0514937	−	0.998673i	$-0.483602\pi$
$751$	1.59178e6	0.102987	0.0514937	+	0.998673i	$0.483602\pi$
$752$	0	0
$753$	−2.00852e6	−0.129089
$754$	0	0
$755$	1.98376e7	1.26655
$756$	0	0
$757$	−2.50784e7	−1.59060	−0.795298	−	0.606219i	$-0.792685\pi$
$757$	−2.50784e7	−1.59060	−0.795298	+	0.606219i	$0.792685\pi$
$758$	0	0
$759$	−5.72831e7	−3.60929
$760$	0	0
$761$	8.26521e6	0.517359	0.258680	−	0.965963i	$-0.416713\pi$
$761$	8.26521e6	0.517359	0.258680	+	0.965963i	$0.416713\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.21440e6	−0.0750252
$766$	0	0
$767$	1.75045e6	0.107439
$768$	0	0
$769$	−2.49439e7	−1.52107	−0.760535	−	0.649297i	$-0.775063\pi$
$769$	−2.49439e7	−1.52107	−0.760535	+	0.649297i	$0.775063\pi$
$770$	0	0
$771$	−9.63099e6	−0.583492
$772$	0	0
$773$	−2.58183e7	−1.55410	−0.777049	−	0.629440i	$-0.783284\pi$
$773$	−2.58183e7	−1.55410	−0.777049	+	0.629440i	$0.783284\pi$
$774$	0	0
$775$	3.00896e6	0.179954
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.36489e7	−1.39626
$780$	0	0
$781$	−3.09758e7	−1.81717
$782$	0	0
$783$	−8.49394e7	−4.95113
$784$	0	0
$785$	−7.66295e6	−0.443836
$786$	0	0
$787$	−1.16838e7	−0.672431	−0.336216	−	0.941785i	$-0.609147\pi$
$787$	−1.16838e7	−0.672431	−0.336216	+	0.941785i	$0.609147\pi$
$788$	0	0
$789$	−2.59181e7	−1.48221
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4.71980e6	0.266526
$794$	0	0
$795$	−1.69967e7	−0.953779
$796$	0	0
$797$	1.49383e7	0.833019	0.416509	−	0.909131i	$-0.363253\pi$
$797$	1.49383e7	0.833019	0.416509	+	0.909131i	$0.363253\pi$
$798$	0	0
$799$	651691.	0.0361139
$800$	0	0
$801$	8.84197e7	4.86931
$802$	0	0
$803$	−3.00492e7	−1.64454
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.93364e7	−2.66676
$808$	0	0
$809$	1.78022e7	0.956320	0.478160	−	0.878273i	$-0.341304\pi$
$809$	1.78022e7	0.956320	0.478160	+	0.878273i	$0.341304\pi$
$810$	0	0
$811$	1.49528e7	0.798307	0.399154	−	0.916884i	$-0.369304\pi$
$811$	1.49528e7	0.798307	0.399154	+	0.916884i	$0.369304\pi$
$812$	0	0
$813$	−2.01695e6	−0.107021
$814$	0	0
$815$	−1.12304e7	−0.592247
$816$	0	0
$817$	2.63760e7	1.38246
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.78509e7	−1.44205	−0.721027	−	0.692907i	$-0.756330\pi$
$821$	−2.78509e7	−1.44205	−0.721027	+	0.692907i	$0.756330\pi$
$822$	0	0
$823$	−56865.0	−0.00292648	−0.00146324	−	0.999999i	$-0.500466\pi$
$823$	−56865.0	−0.00292648	−0.00146324	+	0.999999i	$0.500466\pi$
$824$	0	0
$825$	−1.56210e7	−0.799049
$826$	0	0
$827$	1.32146e7	0.671878	0.335939	−	0.941884i	$-0.390946\pi$
$827$	1.32146e7	0.671878	0.335939	+	0.941884i	$0.390946\pi$
$828$	0	0
$829$	−2.67763e7	−1.35321	−0.676604	−	0.736347i	$-0.736549\pi$
$829$	−2.67763e7	−1.35321	−0.676604	+	0.736347i	$0.736549\pi$
$830$	0	0
$831$	1.61834e7	0.812957
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3.09943e7	1.53839
$836$	0	0
$837$	−3.34658e7	−1.65116
$838$	0	0
$839$	2.78838e7	1.36756	0.683781	−	0.729687i	$-0.260334\pi$
$839$	2.78838e7	1.36756	0.683781	+	0.729687i	$0.260334\pi$
$840$	0	0
$841$	4.96852e7	2.42235
$842$	0	0
$843$	2.42316e7	1.17439
$844$	0	0
$845$	1.16580e7	0.561671
$846$	0	0
$847$	0	0
$848$	0	0
$849$	5.30842e7	2.52753
$850$	0	0
$851$	−2.71740e6	−0.128626
$852$	0	0
$853$	2.57318e7	1.21087	0.605436	−	0.795894i	$-0.292999\pi$
$853$	2.57318e7	1.21087	0.605436	+	0.795894i	$0.292999\pi$
$854$	0	0
$855$	−1.02287e8	−4.78524
$856$	0	0
$857$	−8.30602e6	−0.386315	−0.193157	−	0.981168i	$-0.561873\pi$
$857$	−8.30602e6	−0.386315	−0.193157	+	0.981168i	$0.561873\pi$
$858$	0	0
$859$	2.15556e7	0.996731	0.498365	−	0.866967i	$-0.333934\pi$
$859$	2.15556e7	0.996731	0.498365	+	0.866967i	$0.333934\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.12531e7	0.514334	0.257167	−	0.966367i	$-0.417211\pi$
$863$	1.12531e7	0.514334	0.257167	+	0.966367i	$0.417211\pi$
$864$	0	0
$865$	4.93832e6	0.224408
$866$	0	0
$867$	4.10360e7	1.85403
$868$	0	0
$869$	4.73285e7	2.12605
$870$	0	0
$871$	2.28664e7	1.02130
$872$	0	0
$873$	−1.75692e7	−0.780220
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.42464e7	1.50354	0.751771	−	0.659424i	$-0.229200\pi$
$877$	3.42464e7	1.50354	0.751771	+	0.659424i	$0.229200\pi$
$878$	0	0
$879$	−7.08637e6	−0.309351
$880$	0	0
$881$	−6.79016e6	−0.294741	−0.147370	−	0.989081i	$-0.547081\pi$
$881$	−6.79016e6	−0.294741	−0.147370	+	0.989081i	$0.547081\pi$
$882$	0	0
$883$	3.66001e7	1.57972	0.789861	−	0.613285i	$-0.210152\pi$
$883$	3.66001e7	1.57972	0.789861	+	0.613285i	$0.210152\pi$
$884$	0	0
$885$	7.42238e6	0.318555
$886$	0	0
$887$	1.33180e7	0.568369	0.284185	−	0.958770i	$-0.408277\pi$
$887$	1.33180e7	0.568369	0.284185	+	0.958770i	$0.408277\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	8.82809e7	3.72539
$892$	0	0
$893$	5.48908e7	2.30341
$894$	0	0
$895$	−1.95506e7	−0.815836
$896$	0	0
$897$	−4.18958e7	−1.73856
$898$	0	0
$899$	2.76571e7	1.14132
$900$	0	0
$901$	−297883.	−0.0122246
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.33923e7	−0.543544
$906$	0	0
$907$	1.70149e7	0.686769	0.343385	−	0.939195i	$-0.388427\pi$
$907$	1.70149e7	0.686769	0.343385	+	0.939195i	$0.388427\pi$
$908$	0	0
$909$	6.29861e7	2.52834
$910$	0	0
$911$	8.07087e6	0.322199	0.161100	−	0.986938i	$-0.448496\pi$
$911$	8.07087e6	0.322199	0.161100	+	0.986938i	$0.448496\pi$
$912$	0	0
$913$	−2.75474e7	−1.09371
$914$	0	0
$915$	2.00132e7	0.790250
$916$	0	0
$917$	0	0
$918$	0	0
$919$	3.58766e7	1.40127	0.700637	−	0.713518i	$-0.252899\pi$
$919$	3.58766e7	1.40127	0.700637	+	0.713518i	$0.252899\pi$
$920$	0	0
$921$	−1.14696e7	−0.445553
$922$	0	0
$923$	−2.26552e7	−0.875312
$924$	0	0
$925$	−741029.	−0.0284761
$926$	0	0
$927$	1.58604e7	0.606198
$928$	0	0
$929$	1.86363e7	0.708468	0.354234	−	0.935157i	$-0.384742\pi$
$929$	1.86363e7	0.708468	0.354234	+	0.935157i	$0.384742\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	6.19178e7	2.32869
$934$	0	0
$935$	−1.21235e6	−0.0453523
$936$	0	0
$937$	1.78637e7	0.664697	0.332348	−	0.943157i	$-0.392159\pi$
$937$	1.78637e7	0.664697	0.332348	+	0.943157i	$0.392159\pi$
$938$	0	0
$939$	9.38371e7	3.47305
$940$	0	0
$941$	−3.37739e7	−1.24339	−0.621695	−	0.783260i	$-0.713556\pi$
$941$	−3.37739e7	−1.24339	−0.621695	+	0.783260i	$0.713556\pi$
$942$	0	0
$943$	−2.91418e7	−1.06718
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.53291e7	−1.28014	−0.640070	−	0.768317i	$-0.721095\pi$
$947$	−3.53291e7	−1.28014	−0.640070	+	0.768317i	$0.721095\pi$
$948$	0	0
$949$	−2.19774e7	−0.792157
$950$	0	0
$951$	7.62856e7	2.73522
$952$	0	0
$953$	−3.03334e7	−1.08190	−0.540952	−	0.841053i	$-0.681936\pi$
$953$	−3.03334e7	−1.08190	−0.540952	+	0.841053i	$0.681936\pi$
$954$	0	0
$955$	3.77005e7	1.33764
$956$	0	0
$957$	−1.43582e8	−5.06780
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.77323e7	−0.619380
$962$	0	0
$963$	6.44262e7	2.23870
$964$	0	0
$965$	2.34984e7	0.812307
$966$	0	0
$967$	−4.48744e7	−1.54323	−0.771617	−	0.636087i	$-0.780552\pi$
$967$	−4.48744e7	−1.54323	−0.771617	+	0.636087i	$0.780552\pi$
$968$	0	0
$969$	−2.52662e6	−0.0864430
$970$	0	0
$971$	2.01491e7	0.685817	0.342909	−	0.939369i	$-0.388588\pi$
$971$	2.01491e7	0.685817	0.342909	+	0.939369i	$0.388588\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−1.14249e7	−0.384894
$976$	0	0
$977$	4.29285e7	1.43883	0.719415	−	0.694580i	$-0.244410\pi$
$977$	4.29285e7	1.43883	0.719415	+	0.694580i	$0.244410\pi$
$978$	0	0
$979$	8.82707e7	2.94347
$980$	0	0
$981$	−4.55714e7	−1.51189
$982$	0	0
$983$	4.91728e7	1.62308	0.811542	−	0.584294i	$-0.198629\pi$
$983$	4.91728e7	1.62308	0.811542	+	0.584294i	$0.198629\pi$
$984$	0	0
$985$	2.91193e7	0.956292
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.25023e7	1.05663
$990$	0	0
$991$	3.92457e7	1.26943	0.634714	−	0.772748i	$-0.281118\pi$
$991$	3.92457e7	1.26943	0.634714	+	0.772748i	$0.281118\pi$
$992$	0	0
$993$	5.30536e7	1.70742
$994$	0	0
$995$	2.82216e7	0.903698
$996$	0	0
$997$	5.79147e7	1.84523	0.922616	−	0.385721i	$-0.126047\pi$
$997$	5.79147e7	1.84523	0.922616	+	0.385721i	$0.126047\pi$
$998$	0	0
$999$	8.24177e6	0.261280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.bh.1.1		4
4.3	odd	2		392.6.a.h.1.4	yes	4
7.6	odd	2	inner	784.6.a.bh.1.4		4
28.3	even	6		392.6.i.m.177.4		8
28.11	odd	6		392.6.i.m.177.1		8
28.19	even	6		392.6.i.m.361.4		8
28.23	odd	6		392.6.i.m.361.1		8
28.27	even	2		392.6.a.h.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.6.a.h.1.1	✓	4	28.27	even	2
392.6.a.h.1.4	yes	4	4.3	odd	2
392.6.i.m.177.1		8	28.11	odd	6
392.6.i.m.177.4		8	28.3	even	6
392.6.i.m.361.1		8	28.23	odd	6
392.6.i.m.361.4		8	28.19	even	6
784.6.a.bh.1.1		4	1.1	even	1	trivial
784.6.a.bh.1.4		4	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q-28.9227 q^{3} -63.5336 q^{5} +593.520 q^{9} +O(q^{10})\)	\(q-28.9227 q^{3} -63.5336 q^{5} +593.520 q^{9} +592.520 q^{11} +433.359 q^{13} +1837.56 q^{15} +32.2049 q^{17} +2712.56 q^{19} +3342.60 q^{23} +911.520 q^{25} -10138.0 q^{27} +8378.33 q^{29} +3301.03 q^{31} -17137.3 q^{33} -812.959 q^{37} -12533.9 q^{39} -8718.28 q^{41} +9723.64 q^{43} -37708.5 q^{45} +20235.8 q^{47} -931.451 q^{51} -9249.61 q^{53} -37645.0 q^{55} -78454.5 q^{57} +4039.25 q^{59} +10891.2 q^{61} -27532.8 q^{65} +52765.6 q^{67} -96677.0 q^{69} -52278.1 q^{71} -50714.2 q^{73} -26363.6 q^{75} +79876.5 q^{79} +148992. q^{81} -46491.8 q^{83} -2046.09 q^{85} -242324. q^{87} +148975. q^{89} -95474.7 q^{93} -172339. q^{95} -29601.7 q^{97} +351673. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 836 q^{9}+O(q^{10}) \) 4 * q + 836 * q^9	\( 4 q + 836 q^{9} + 832 q^{11} + 2736 q^{15} + 5680 q^{23} + 2108 q^{25} + 8904 q^{29} - 6328 q^{37} - 20912 q^{39} + 28128 q^{43} + 30112 q^{51} + 9144 q^{53} - 164624 q^{57} - 83984 q^{65} + 52640 q^{67} + 21600 q^{71} + 282592 q^{79} + 326804 q^{81} - 226592 q^{85} - 255776 q^{93} - 290992 q^{95} + 765312 q^{99}+O(q^{100}) \) 4 * q + 836 * q^9 + 832 * q^11 + 2736 * q^15 + 5680 * q^23 + 2108 * q^25 + 8904 * q^29 - 6328 * q^37 - 20912 * q^39 + 28128 * q^43 + 30112 * q^51 + 9144 * q^53 - 164624 * q^57 - 83984 * q^65 + 52640 * q^67 + 21600 * q^71 + 282592 * q^79 + 326804 * q^81 - 226592 * q^85 - 255776 * q^93 - 290992 * q^95 + 765312 * q^99

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